Chemistry Formulas Compendium

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### Atomic Structure - **Estimation of closest distance of approach (derivation of $\alpha$-particle):** $R = \frac{4KZe^2}{m_0V_0^2}$ - **Radius of a nucleus:** $R = R_0 (A)^{1/3}$ cm - **Planck's Quantum Theory:** Energy of one photon = $h\nu = \frac{hc}{\lambda}$ - **Photoelectric Effect:** $h\nu = h\nu_0 + \frac{1}{2} m_e v^2$ - **Bohr's Model for Hydrogen like atoms:** 1. $mvr = n \frac{h}{2\pi}$ (Quantization of angular momentum) 2. $E_n = -\frac{Z^2}{n^2} \times 2.18 \times 10^{-18}$ J/atom $= -13.6 \frac{Z^2}{n^2}$ eV ; $E_1 = -2\pi^2 me^4 / n^2$ 3. $r_n = \frac{n^2}{Z} \times \frac{h^2}{4\pi^2 e^2m} = 0.529 \times \frac{n^2}{Z}$ Å 4. $v = \frac{2\pi Ze^2}{nh} = 2.18 \times 10^8 \times \frac{Z}{n}$ m/s - **De-Broglie wavelength:** - $\lambda = \frac{h}{mv} = \frac{h}{p}$ (for photon) - **Wavelength of emitted photon:** - $\frac{1}{\lambda} = \nu = RZ^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$ - **No. of photons emitted by a sample of H atom:** - $\frac{\Delta n (\Delta n + 1)}{2}$ - **Heisenberg's uncertainty principle:** - $\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$ or $\Delta x \cdot m \Delta v \ge \frac{h}{4\pi}$ or $\Delta x \cdot \Delta v \ge \frac{h}{4\pi m}$ - **Quantum Numbers:** - Principal quantum number ($n$) = 1, 2, 3, 4 ... to $\infty$. - Orbital angular momentum of electron in any orbit = $\frac{nh}{2\pi}$ - Azimuthal quantum number ($l$) = 0, 1, ..., to ($n-1$) - Number of orbitals in a subshell = $2l+1$ - Maximum number of electrons in particular subshell = $2 \times (2l+1)$ - Orbital angular momentum $L = \frac{h}{2\pi} \sqrt{l(l+1)}$ = $\hbar \sqrt{l(l+1)}$ where $\hbar = \frac{h}{2\pi}$ ### Stoichiometry - **Relative atomic mass (R.A.M) =** $\frac{\text{Mass of one atom of an element}}{1/12 \times \text{mass of one carbon atom}}$ = Total Number of nucleons - **Y-map (Mole Concept Diagram):** - (Diagram showing interconversion between Number, Mole, Mass, Volume at STP) - Number $\leftrightarrow$ Mole (via Avogadro's number $N_A$) - Mole $\leftrightarrow$ Mass (via mol. wt. / At. wt.) - Mole $\leftrightarrow$ Volume at STP (via $22.4$ L) - **Density:** - Specific gravity = $\frac{\text{density of the substance}}{\text{density of water at } 4^\circ C}$ - **For gases:** - Absolute density (mass/volume) = $\frac{\text{Molar mass of the gas}}{\text{Molar volume of the gas}} \Rightarrow p = \frac{PM}{RT}$ - Vapour density = V.D. = $\frac{d_{\text{gas}}}{d_{H_2}} = \frac{PM_{\text{gas}}RT}{PM_{H_2}RT} = \frac{M_{\text{gas}}}{M_{H_2}} = \frac{M_{\text{gas}}}{2}$ - $M_{\text{gas}} = 2 \times \text{V.D.}$ - **Mole-mole analysis:** (Flowchart for mole-mole relationships in reactions) - **Concentration terms:** - **Molarity (M):** - Molality (m) = $\frac{w \times 1000}{(\text{Mol. wt of solute}) \times V_{\text{ml}}}$ - Molarity = $\frac{\text{number of moles of solute}}{\text{mass of solvent in gram}} \times 1000 = \frac{w}{M_1W_2} \times 1000$ (where $w$ is mass of solute, $M_1$ is molar mass of solute, $W_2$ is mass of solvent in grams) - **Mole fraction (x):** - Mole fraction of solution $(x_1) = \frac{n}{n+N}$ ; Mole fraction of solvent $(x_2) = \frac{N}{n+N}$ - $x_1 + x_2 = 1$ - **% Calculation:** - (i) % w/w = $\frac{\text{mass of solute in gm}}{\text{mass of solution in gm}} \times 100$ - (ii) % w/v = $\frac{\text{mass of solute in gm}}{\text{mass of solution in ml}} \times 100$ - (iii) % v/v = $\frac{\text{Volume of solution in ml}}{\text{Volume of solution}} \times 100$ - **Derive the following conversion:** 1. Mole fraction of solute into molarity of solution M = $\frac{x_1 \rho \times 1000}{x_1 M_1 + M_2 x_2}$ 2. Molarity into mole fraction $x_2 = \frac{MM_1 \times 1000}{\rho \times 1000 - MM_1}$ 3. Mole fraction into molality $m = \frac{X_2 \times 1000}{x_1 M_1}$ 4. Molality into mole fraction $x_2 = \frac{mM_1}{1000 + mM_1}$ 5. Molality into molarity M = $\frac{m \rho \times 1000}{1000 + mM_2}$ 6. Molarity into Molality M = $\frac{M \times 1000}{1000 \rho - MM_2}$ (M and $M_2$ are molar masses of solvent and solute. $\rho$ is density of solution (gm/mL). M = Molarity (mole/lit.), m = Molality (mole/kg). $x_1$ = Mole fraction of solvent, $x_2$ = Mole fraction of solute) - **Average/Mean atomic mass:** - $A_{\text{avg}} = \frac{A_1 x_1 + A_2 x_2 + ... + A_n x_n}{100}$ - **Mean molar mass or molecular mass:** - $M_{\text{avg}} = \frac{\sum n_i M_i}{\sum n_i}$ or $\frac{\sum_i M_i}{\sum_i n_i}$ - **Calculation of individual oxidation number:** - **Formula : Oxidation Number =** number of electrons in the valence shell - number of electrons left after bonding - **Concept of Equivalent weight/Mass:** - For elements, equivalent weight (E) = $\frac{\text{Atomic weight}}{\text{Valency-factor}}$ - For acid/base, E = $\frac{M}{\text{Basicity / Acidity}}$ Where M = Molar mass - For O.A./R.A, E = $\frac{M}{\text{no. of moles of } e^- \text{gained/lost}}$ - Equivalent weight (E) = $\frac{\text{Atomic or molecular weight}}{\text{v.f.}}$ (v.f. = valency factor) - **Concept of number of equivalents:** - Wt / W = Wt / E = M/n - No. of equivalents of solute = Wt / E - No. of equivalents of solute = No. of moles of solute $\times$ v.f. - **Normality (N):** - Normality (N) = $\frac{\text{Number of equivalents of solute}}{\text{Volume of solution (in litres)}}$ - Normality = Molarity $\times$ v.f. - **Calculation of valency Factor:** - n-factor of acid = basicity = no. of H$^+$ ion(s) furnished per molecule of the acid. - n-factor of base = acidity = no. of OH$^-$ ion(s) furnished by the base per molecule. - **At equivalence point:** - $N_1 V_1 = N_2 V_2$ - $n_1 M_1 V_1 = n_2 M_2 V_2$ - **Volume strength of H$_2$O$_2$:** - 20V H$_2$O$_2$ means one litre of this sample of H$_2$O$_2$ on decomposition gives 20 lit. of O$_2$ gas at S.T.P. - Normality of H$_2$O$_2$ (N) = $\frac{\text{Volume strength of H}_2\text{O}_2}{5.6}$ - Molarity of H$_2$O$_2$ (M) = $\frac{\text{Volume strength of H}_2\text{O}_2}{11.2}$ - **Measurement of Hardness:** - Hardness in ppm = $\frac{\text{mass of CaCO}_3}{\text{Total mass of water}} \times 10^6$ - **Calculation of available chlorine from a sample of bleaching powder:** - % of Cl$_2$ = $\frac{3.55 \times V \times (\text{ml})}{W(\text{g})}$ where x = molarity of hypo solution and V = mL. of hypo solution used in titration. ### Gaseous State - **Temperature Scale:** - $\frac{C - 0}{100 - 0} = \frac{K - 273}{373 - 273} = \frac{F - 32}{212 - 32}$ R-R(0) / R(100)-R(0) where R = Temp. on unknown scale. - **Boyle's law and measurement of pressure:** - $V \propto \frac{1}{P}$ - $P_1V_1 = P_2V_2$ - **Charles law:** - At constant pressure, $V \propto T$ or $\frac{V_1}{T_1} = \frac{V_2}{T_2}$ - **Gay-lussac's law:** - At constant volume, $P \propto T$ - $\frac{P_1}{T_1} = \frac{P_2}{T_2} \rightarrow$ temp on absolute scale - **Ideal Gas Equation:** - $PV = nRT$ - $PV = \frac{w}{m} RT$ or $P = \frac{d}{m} RT$ or $Pm = dRT$ - **Dalton's law of partial pressure:** - $P_1 = \frac{n_1 RT}{V}$ , $P_2 = \frac{n_2 RT}{V}$ , $P_3 = \frac{n_3 RT}{V}$ and so on. - Total pressure $= P_1 + P_2 + P_3 + ......$ - **Amagat's law of partial volume:** - $V = V_1 + V_2 + V_3 + ......$ - **Average molecular mass of gaseous mixture:** - $M_{\text{mix}} = \frac{\sum n_i M_i}{\sum n_i} = \frac{\text{Total mass of mixture}}{\text{Total no. of moles in mixture}}$ - **Graham's Law:** - Rate of diffusion $r \propto \frac{1}{\sqrt{d}}$ ; $d = \text{density of gas}$ - $\frac{r_1}{r_2} = \sqrt{\frac{d_2}{d_1}} = \sqrt{\frac{M_2}{M_1}} = \sqrt{\frac{V_2 D_1}{V_1 D_2}}$ - **Kinetic Theory of Gases:** - $PV = \frac{1}{3} mN U^2$ Kinetic equation of gases - Average K.E. for one mole $= N_A \left(\frac{1}{2} m U^2\right) = \frac{3}{2} KN_A T = \frac{3}{2} RT$ - Root mean square speed $U_{\text{rms}} = \sqrt{\frac{3RT}{M}}$ molar mass must be in kg/mole. - Average speed $U_{\text{avg}} = U_1 + U_2 + ... + U_n$ $U_{\text{avg}} = \sqrt{\frac{8RT}{\pi M}}$ - Most probable speed $U_{\text{MPS}} = \sqrt{\frac{2RT}{M}}$ - **Vander wall's equation:** - $\left(P + \frac{an^2}{V^2}\right) (V - nb) = nRT$ - **Critical constants:** - $V_c = 3b$, $P_c = \frac{a}{27b^2}$, $T_c = \frac{8a}{27Rb}$ - **Vander wall equation in virial form:** - $Z = 1 + \frac{b}{V_m} + \frac{b^2}{V_m^2} + \frac{b^3}{V_m^3} ... - \frac{a}{V_m RT} = 1 + \frac{1}{V_m} \left(b - \frac{a}{RT}\right) + \frac{b^2}{V_m^2} + \frac{b^3}{V_m^3} + ........$ - **Reduced Equation of state:** - $\left(P_r + \frac{3}{V_r^2}\right) (3V_r - 1) = 8 T_r$ ### Thermodynamics - **Thermodynamic processes:** 1. **Isothermal process:** $T = \text{constant}$, $dT = 0$, $\Delta T = 0$ 2. **Isochoric process:** $V = \text{constant}$, $dV = 0$, $\Delta V = 0$ 3. **Isobaric process:** $P = \text{constant}$, $dP = 0$, $\Delta P = 0$ 4. **Adiabatic process:** $q = 0$ or heat exchange with the surrounding = 0(zero) - **IUPAC Sign convention about Heat and Work:** - Work done on the system = Positive - Work done by the system = Negative - **1st Law of Thermodynamics:** - $\Delta U = (U_2 - U_1) = q + w$ - **Law of equipartition of energy:** - $U = \frac{f}{2} nRT$ (only for ideal gas) - $\Delta E = \frac{f}{2} nR (\Delta T)$ - where $f = \text{degrees of freedom for that gas.}$ (Translational + Rotational) - $f = 3$ for monoatomic - $f = 5$ for diatomic or linear polyatomic - $f = 6$ for non - linear polyatomic - **Calculation of heat ($q$):** - Total heat capacity: $C_p = \frac{\Delta q}{\Delta T} = \frac{dq}{dT}$ $J/\text{°C}$ - Molar heat capacity: $C = \frac{\Delta q}{n \Delta T} = \frac{dq}{ndT}$ J mole$^{-1}$ K$^{-1}$ - $C_p = C_v + R / (\gamma - 1)$ - Specific heat capacity (s): $S = \frac{\Delta q}{m \Delta T} = \frac{dq}{mdT}$ Jgm$^{-1}$ K$^{-1}$ - **WORK DONE ($w$):** - **Isothermal Reversible expansion/compression of an ideal gas:** $W = -nRT \ln (V_2/V_1)$ - **Reversible and irreversible isochoric processes:** - Since $dV = 0$ - So $dW = -P_{\text{ext}} dV = 0$. - **Reversible isobaric process:** $W = P (V_1 - V_2)$ - **Adiabatic reversible expansion:** - $T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1}$ - **Reversible Work:** $W = \frac{P_2 V_2 - P_1 V_1}{\gamma - 1} = \frac{nR (T_2 - T_1)}{\gamma - 1}$ - **Irreversible Work:** $W = \frac{P_2 V_2 - P_1 V_1}{\gamma - 1} = \frac{nR (T_2 - T_1)}{\gamma - 1}$ $nC_v (T_2 - T_1) = -P_{\text{ext}} (V_2 - V_1)$ and use $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$ - **Free expansion** - Always going to be irreversible and since $P_{\text{ext}} = 0$ - So $dW = -P_{\text{ext}} dV = 0$ - If no heat is supplied $q = \Delta T = 0$. - then $\Delta U = 0$ so $\Delta W = -P \Delta V$ - **Application of 1st Law:** - $\Delta U = \Delta Q + \Delta W$ - $\Delta U = \Delta Q - P \Delta V$ - **Constant volume process:** - Heat given at constant volume = change in internal energy - $dU = (dq)_v$ - $du = nC_v dT$ - $C_v = \frac{1}{n} \frac{dU}{dT} = \frac{f}{2}R$ - **Constant pressure process:** - H = Enthalpy (state function and extensive property) - $H = U + PV$ - $C_p - C_v = R$ (only for ideal gas) - **Second Law Of Thermodynamics:** - $\Delta S_{\text{universe}} = \Delta S_{\text{system}} + \Delta S_{\text{surrounding}} > 0$ for a spontaneous process. - **Entropy ($S$):** - $\Delta S = \int \frac{dq_{\text{rev}}}{T}$ - $\Delta S_{\text{system}} = \frac{q_{\text{rev}}}{T}$ - **Entropy calculation for an ideal gas undergoin a process:** - State A $\xrightarrow{\text{irr}}$ State B - $P_1, V_1, T_1 \xrightarrow{\Delta S_{\text{system}}} P_2, V_2, T_2$ - $\Delta S_{\text{system}} = nC_v \ln \frac{T_2}{T_1} + nR \ln \frac{V_2}{V_1}$ (only for an ideal gas) - **Third Law of Thermodynamics:** - The entropy of perfect crystals of all pure elements & compounds is zero at the absolute zero of temperature. - **Gibb's free energy ($G$):** (State function and an extensive property) - $G_{\text{system}} = H_{\text{system}} - TS_{\text{system}}$ - **Criteria of spontaneity:** - (i) If $\Delta G_{\text{system}}$ is (-ve) $ 0 \Rightarrow$ system is at equilibrium. - **Physical interpretation of $\Delta G$:** - The maximum amount of non-expansional (compression) work which can be performed. - $\Delta G = dw_{\text{non-exp}} = dH - TdS$. - **Standard Free Energy Change ($\Delta G^\circ$):** 1. $\Delta G^\circ = -2.303 RT \log_{10} K$ 2. At equilibrium $\Delta G = 0$. 3. The decrease in free energy $(-\Delta G)$ is given as: $\Delta G = W_{\text{net}} = -2.303 nRT \log_{10} \frac{V_2}{V_1}$ 4. $\Delta G_{\text{f}}^\circ$ for elemental state = 0 5. $\Delta G_{\text{r}}^\circ = \sum G_{\text{products}}^\circ - \sum G_{\text{Reactants}}^\circ$ - **Thermochemistry:** - Change in standard enthalpy $\Delta H^\circ = H_{m,2}^\circ - H_{m,1}^\circ$ - $\rightarrow$ = heat added at constant pressure. = $C_p \Delta T$. - $\rightarrow$ $H_{\text{products}} > H_{\text{reactants}}$ - Reaction should be endothermic as we have to give extra heat to reactants to get these converted into products - and if $H_{\text{products}} ### Chemical Equilibrium - **At equilibrium:** - (i) Rate of forward reaction = rate of backward reaction - (ii) Concentration (mole/litre) of reactant and product becomes constant. - (iii) $\Delta G = 0$. - (iv) $Q = K_{\text{eq}}$ - **Equilibrium constant ($K$):** - $K = \frac{\text{rate constant of forward reaction}}{\text{rate constant of backward reaction}} = \frac{K_{\text{f}}}{K_{\text{b}}}$ - **Equilibrium constant in terms of concentration ($K_c$):** - $K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b}$ - **Equilibrium constant in terms of partial pressure ($K_p$):** - $K_p = \frac{[P_C]^c [P_D]^d}{[P_A]^a [P_B]^b}$ - **Equilibrium constant in terms of mole fraction ($K_x$):** - $K_x = \frac{x_C^c x_D^d}{x_A^a x_B^b}$ - **Relation between $K_p$ & $K_c$:** - $K_p = K_c (RT)^{\Delta n}$ - **Relation between $K_p$ & $K_x$:** - $K_p = K_x (P)^{\Delta n}$ - **Thermodynamics of Equilibrium:** - $\Delta G = \Delta G^\circ + RT \ln Q$ - $\log \frac{K_2}{K_1} = \frac{\Delta H}{2.303 R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)$ ; $\Delta H = \text{Enthalpy of reaction}$ - **Relation between equilibrium constant & standard free energy change:** - $\Delta G^\circ = -2.303 RT \log K$ - **Reaction Quotient ($Q$):** - $Q = \frac{[C]^c [D]^d}{[A]^a [B]^b}$ - **Degree of Dissociation ($\alpha$):** - $\alpha = \frac{\text{no. of moles dissociated}}{\text{initial no. of moles taken}} = \frac{\text{fraction of moles dissociated out of 1 mole.}}{\text{no. of moles initially taken}}$ - Note : % dissociation = $\alpha \times 100$ - **Observed molecular weight and Observed Vapour Density of the mixture:** - Observed molecular weight of $A_n(g) = \frac{\text{molecular weight of equilibrium mixture}}{\text{total no. of moles}}$ - $\alpha = \frac{D - d}{(n-1)d}$ where $D = \frac{M_1 - M_2}{(n-1)M_0}$ - **External factor affecting equilibrium:** - **Le Chatelier's Principle:** If a system at equilibrium is subjected to a disturbance or stress that changes any of the factors that determine the state of equilibrium, the system will react in such a way as to minimize the effect of the disturbance. - **Effect of concentration:** - If the concentration of reactant is increased at equilibrium then reaction shift in the forward direction. - If the concentration of product is increased then equilibrium shifts in the backward direction - **Effect of volume:** - If volume is increased pressure decreases hence reaction will shift in the direction in which pressure increases that is in the direction in which number of moles of gases increases and vice versa. - if volume is increased then for $\Delta n > 0$ reaction will shift in the forward direction - $\Delta n 0$ reaction will shift in the forward direction - $\Delta n 0$) reaction value of the equilibrium constant increases with the rise in temperature - For exothermic ($\Delta H 0$, reaction shifts in the forward direction with increase in temperature - For $\Delta H ### Ionic Equilibrium - **Vant Hoff equation:** $\log \frac{K_2}{K_1} = \frac{\Delta H}{2.303R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)$ - **Ostwald Dilution Law:** 1. **Dissociation constant of weak acid ($K_a$):** $K_a = \frac{[H^+][A^-]}{[HA]} = \frac{C\alpha^2}{1-\alpha}$ - If $\alpha 14$] - pH = $-\log [H^+]$ ; $[H^+] = 10^{-\text{pH}}$ - pOH = $-\log [OH^-]$ ; $[OH^-] = 10^{-\text{pOH}}$ - pKa = $-\log K_a$ ; $K_a = 10^{-\text{pKa}}$ - pKb = $-\log K_b$ ; $K_b = 10^{-\text{pKb}}$ - **Properties of water:** 1. In pure water $[H^+] = [OH^-]$ so it is Neutral. 2. Molar concentration / Molarity of water = 55.56 M. 3. **Ionic product of water ($K_w$):** - $K_w = [H^+][OH^-] = 10^{-14}$ at $25^\circ C$ (experimentally) - pH $ 7$ or pOH $> 7 \Rightarrow$ Basic 4. **Degree of dissociation of water:** $\alpha = \frac{\text{no. of moles dissociated}}{\text{Total No. of moles initially taken}} = \frac{10^{-7}}{55.55} = 18 \times 10^{-10}$ or $1.8 \times 10^{-7}$% 5. **Absolute dissociation constant of water:** - $K_w = [H^+][OH^-] = 10^{-7} \times 10^{-7}$ - $K_a = K_w = \frac{[H^+][OH^-]}{[H_2O]} = \frac{10^{-7} \times 10^{-7}}{55.55} = 1.8 \times 10^{-16}$ - **pH Calculations of Different Types of Solutions:** - (a) **Strong acid solution :** - (i) If concentration is greater than $10^{-6}$ M - In this case H$^+$ ions coming from water can be neglected, - if concentration is less than $10^{-6}$ M - (ii) In this case H$^+$ ions coming from water cannot be neglected - (b) **Strong base solution :** - Using similar method as in part (a) calculate first [OH$^-$] and then use $[H^+] \times [OH^-] = 10^{-14}$ - (c) **pH of mixture of two strong acids :** - Number of H$^+$ ions from I-solution = $N_1 V_1$ - Number of H$^+$ ions from II-solution = $N_2 V_2$ - $[H^+] = \frac{N_1 V_1 + N_2 V_2}{V_1 + V_2}$ - (d) **pH of mixture of two strong bases :** - $[OH^-] = N = \frac{N_1 V_1 + N_2 V_2}{V_1 + V_2}$ - (e) **pH of mixture of a strong acid and a strong base :** - If $N_1 V_1 > N_2 V_2$, then solution will be acidic in nature and $[H^+] = N = \frac{N_1 V_1 - N_2 V_2}{V_1 + V_2}$ - If $N_2 V_2 > N_1 V_1$, then solution will be basic in nature and $[OH^-] = N = \frac{N_2 V_2 - N_1 V_1}{V_1 + V_2}$ - (f) **pH of a weak acid(monoprotic) solution :** - $K_a = \frac{[H^+][A^-]}{[HA]} = \frac{C\alpha^2}{1-\alpha}$ - If $\alpha > K_{a_2} >> K_{a_3}$ - pH of H$_2$PO$_4^- = \frac{1}{2} (pK_{a_1} + pK_{a_2})$ - pH of Na$_2$HPO$_4 = \frac{1}{2} (pK_{a_2} + pK_{a_3})$ - pH of Na$_3$PO$_4 = \frac{1}{2} (pK_w + pK_{a_3} + \log C)$ $Sec$ hydrolysis can neglect. - **Buffer Solution :** - (a) **Acidic Buffer :** e.g. CH$_3$COOH and CH$_3$COONa. (weak acid and salt of its conjugate base). - pH = $pK_a + \log \frac{[\text{Salt}]}{[\text{Acid}]}$ [Henderson's equation] - (b) **Basic Buffer :** e.g. NH$_4$OH + NH$_4$Cl. (weak base and salt of its conjugate acid). - pOH = $pK_b + \log \frac{[\text{Salt}]}{[\text{Base}]}$ - **Buffer capacity (index):** - Buffer capacity = $\frac{\text{Total no. of moles of acid / alkali added per litre}}{\text{Change in pH}}$ - Buffer capacity = $\frac{dx}{d(\Delta pH)} = 2.303 \frac{(a+x)(b-x)}{a+b}$ ### Indicator - $HIn \rightleftharpoons H^+ + In^-$ - or $[H^+] = K_{In} \frac{[HIn]}{[In^-]}$ - $\therefore pH = pK_{In} + \log \frac{[HIn]}{[In^-]}$ $\Rightarrow$ pH = $pK_{In} + \log \frac{[\text{ionised form}]}{[\text{Unionised form}]}$ - **Significance of indicators :** - Extent of reaction of different bases with acid (HCl) using two indicators : - **Phenolphthalein** : Methyl Orange - NaOH : 100% reaction is indicated ; 100% reaction is indicated - NH$_4$OH : NaOH + HCl $\rightarrow$ NaCl + H$_2$O ; NaOH + HCl $\rightarrow$ NaCl + H$_2$O - Na$_2$CO$_3$ : 50% reaction using NaHCO$_3$ ; 100% reaction is indicated - stage is indicated - NaHCO$_3$ : Na$_2$CO$_3$ + HCl $\rightarrow$ NaHCO$_3$ + NaCl ; Na$_2$CO$_3$ + 2HCl $\rightarrow$ 2NaCl + H$_2$O + CO$_2$ - No reaction is indicated ; 100% reaction is indicated - NaHCO$_3$ + HCl $\rightarrow$ NaCl + H$_2$O + CO$_2$ - **Isoelectric point :** - $[H^+] = \sqrt{pK_{a_1} K_{a_2}}$ - pH = $\frac{pK_{a_1} + pK_{a_2}}{2}$ - **Solubility product :** - $K_{\text{sp}} = (X)^x (Y)^y = x^x y^y (S)^{x+y}$ - **Condition for precipitation :** - If ionic product $K_{\text{ip}} > K_{\text{sp}}$, precipitation occurs. - If $K_{\text{ip}} = K_{\text{sp}}$ saturated solution (precipitation just begins or is just prevented). ### Electrochemistry - **Electrode Potential** - For any electrode $\rightarrow$ Oxidation potential = - Reduction potential - $E_{\text{cell}} = R.P \text{ of cathode} - R.P \text{ of anode}$ - $E_{\text{cell}} = R.P \text{ of cathode} + O.P \text{ of anode}$ - $E_{\text{cell}}$ is always a +ve quantity & Anode will be electrode of low R.P - $E_{\text{cell}}^\circ = SRP \text{ of cathode} - SRP \text{ of anode.}$ - Greater the SRP value greater will be oxidising power. - **Gibbs Free Energy Change :** - $\Delta G = -nFE_{\text{cell}}$ - $\Delta G^\circ = -nFE_{\text{cell}}^\circ$ - **Nernst Equation : (Effect of concentration and temp of an emf of cell)** - $\Rightarrow \Delta G = \Delta G^\circ + RT \ln Q$ (where Q is raection quotient) - $\Delta G^\circ = -RT \ln K_{\text{eq}}$ - $E_{\text{cell}} = E_{\text{cell}}^\circ - \frac{RT}{nF} \ln Q$ - $E_{\text{cell}} = E_{\text{cell}}^\circ - \frac{2.303 RT}{nF} \log Q$ - $E_{\text{cell}} = E_{\text{cell}}^\circ - \frac{0.0591}{n} \log Q$ [At 298 K] - At chemical equilibrium $\Delta G = 0$ ; $E_{\text{cell}} = 0$. - $E_{\text{cell}}^\circ = \frac{nFE_{\text{cell}}^\circ}{0.0591}$ - $E_{\text{cell}}^\circ = \frac{0.0591}{n} \log K_{\text{eq}}$ - For an electrode M(s)/M$^{n+}$: $E_{M^{n+}/M} = E_{M^{n+}/M}^\circ - \frac{2.303RT}{nF} \log \frac{1}{[M^{n+}]}$ - **Concentration Cell :** A cell in which both the electrodes are made up of same material. - For all concentration cell $E_{\text{cell}}^\circ = 0$. - (a) **Electrolyte Concentration Cell :** - eg. Zn(s) | Zn$^{2+}$(c$_1$) || Zn$^{2+}$(c$_2$) | Zn(s) - $E = \frac{0.0591}{2} \log \frac{C_2}{C_1}$ - (b) **Electrode Concentration Cell :** - eg. Pt, H$_2$(P$_1$ atm) / H$^+$ (1M) / H$_2$ (P$_2$ atm) / Pt - $E = \frac{0.0591}{2} \log \left(\frac{P_1}{P_2}\right)$ - **Different Types of Electrodes :** 1. Metal-Metal Ion Electrode M(s)/M$^{n+}$ : M $\rightleftharpoons$ M$^{n+}$ + ne$^-$ M(s) - $E = E^\circ + \frac{0.0591}{n} \log [M^{n+}]$ 2. Gas-Ion Electrode Pt/H$_2$(P atm)/H$^+$ (XM) - as a reduction electrode H$^+$ (aq) + e$^-$ $\rightleftharpoons$ $\frac{1}{2}$ H$_2$ (P atm) - $E = E^\circ - 0.0591 \log \frac{P_{H_2}^{1/2}}{[H^+]}$ 3. Oxidation-reduction Electrode Pt / Fe$^{3+}$, Fe$^{2+}$ - as a reduction electrode Fe$^{3+}$ + e$^-$ $\rightleftharpoons$ Fe$^{2+}$ - $E = E^\circ - 0.0591 \log \frac{[Fe^{2+}]}{[Fe^{3+}]}$ 4. Metal-Metal insoluble salt Electrode eg. Ag/AgCl, Cl$^-$ - as a reduction electrode AgCl(s) + e$^-$ $\rightleftharpoons$ Ag(s) + Cl$^-$ - $E_{Cl^- / AgCl / Ag} = E_{Cl^- / AgCl / Ag}^\circ - 0.0591 \log [Cl^-]$ - **Calculation of different thermodynamics function of cell reaction** - $\Delta G = -nFE_{\text{cell}}$ - $S = -\left(\frac{d(\Delta G)}{dT}\right)_P$ (At costant pressure). - $\Delta S = -\left[\frac{d(\Delta G)}{dT}\right]_P = nF \left(\frac{dE_{\text{cell}}}{dT}\right)_P$ - $\left(\frac{\partial E}{\partial T}\right)_P =$ Temperature coefficient of e.m.f of the cell. - $E = a + bT + CT^2 + ........$ - $\Delta H = nF \left[T \left(\frac{\partial E}{\partial T}\right)_P - E\right]$ - **$\Delta C_p$ of cell reaction** - $\Delta C_p = \frac{dH}{dT}$ - $\Delta C_p = \frac{d}{dT} (\Delta H)$ - $\Delta C_p = nF T \frac{d^2 E_{\text{cell}}}{dT^2}$ - **Electrolysis :** - (a) $K^+, Ca^{2+}, Na^+, Mg^{2+}, Al^{3+}, Zn^{2+}, Fe^{2+}, H^+, Cu^{2+}, Ag^+, Au^{3+}$. - Increasing order of deposition. - (b) Similarly the anion which is strognger reducing agent(low value of SRP) is liberated first at the anode. - $SO_4^{2-}, NO_3^-, OH^-, Cl^-, Br^-, I^-$ - Increasing order of deposition ### Faraday's Law of Electrolysis - **First Law :** - $w = zq$ $w = z It$ $Z = \text{Eledrochemical equivalent of substance}$ - **Second Law :** - $W \propto E$ - $\frac{W_1}{E_1} = \text{constant}$ $\frac{W_1}{E_1} = \frac{W_2}{E_2} = .........$ - $W = \frac{I \times t \times \text{current efficiency factor}}{96500}$ - **Current Efficiency =** $\frac{\text{actual mass deposited/produced}}{\text{Theoritical mass deposited/produced}} \times 100$ - **Condition for Simultaneous Deposition of Cu & Fe at Cathode** - $E_{\text{Cu}^{2+}/\text{Cu}} = E_{\text{Cu}^{2+}/\text{Cu}}^\circ - \frac{0.0591}{2} \log \frac{1}{[Cu^{2+}]}$ - $E_{\text{Fe}^{2+}/\text{Fe}} = E_{\text{Fe}^{2+}/\text{Fe}}^\circ - \frac{0.0591}{2} \log \frac{1}{[Fe^{2+}]}$ - Condition for the simulataneous deposition of Cu & Fe on cathode. - **Conductance :** - Conductance = $\frac{1}{\text{Resistance}}$ - Specific conductance or conductivity : - (Reciprocal of specific resistance) $K = \frac{1}{\rho}$ $K = \text{specific conductance}$ - Equivalent conductance : - $\lambda_E = \frac{K \times 1000}{\text{Normality}}$ unit : ohm$^{-1}$ cm$^2$ eq$^{-1}$ - Molar conductance : - $\lambda_M = \frac{K \times 1000}{\text{Molarity}}$ unit : ohm$^{-1}$ cm$^2$ mole$^{-1}$ - specific conductance = conductance $\times \frac{l}{a}$ - **Kohlrausch's Law :** - **Variation of $\lambda_M / \lambda_E$ of a solution with concentration :** - (i) **Strong electrolyte** - $\lambda_M = \lambda_M^\circ - b \sqrt{C}$ - (ii) **Weak electrolytes :** $\lambda_M = n_a \lambda_a^\circ + n_c \lambda_c^\circ$ where $\lambda_a$ is the molar conductivity - $n_a =$ No of cations obtained after dissociation per formula unit - $n_c =$ No of anions obtained after dissociation per formula unit - **Application of Kohlrausch Law :** 1. **Calculation of $\lambda_M^\circ$ of weak electrolytes :** - $\lambda_{M(CH_3COOH)}^\circ = \lambda_{M(CH_3COONa)}^\circ + \lambda_{M(HCl)}^\circ - \lambda_{M(NaCl)}^\circ$ 2. To calculate degree of dissociation of a weak electrolyte - $\alpha = \frac{\lambda_M}{\lambda_M^\circ}$ - $K_{eq} = \frac{C\alpha^2}{(1-\alpha)}$ 3. Solubility (S) of sparingly soluble salt & their $K_{\text{sp}}$ - $\lambda_M = \lambda_M^\circ = \frac{K \times 1000}{\text{solubility}}$ - $K_{\text{sp}} = S^2$. - **Ionic Mobility :** It is the distance travelled by the ion per second under the potential gradient of 1 volts per cm. It's unit is cm$^2$ s$^{-1}$ V$^{-1}$. - Absolute ionic mobility: $\lambda_c^\circ = \mu_c$ ; $\lambda_a^\circ = \mu_a$ - $\lambda_c^\circ = F \mu_c^\circ$ ; $\lambda_a^\circ = F \mu_a^\circ$ - Ionic Mobility $\mu = \frac{v}{\text{potential gradient}}$ - **Transport Number :** - $t_c = \frac{\mu_c}{\mu_c + \mu_a}$ ; $t_a = \frac{\mu_a}{\mu_c + \mu_a}$ - Where $t_c =$ Transport Number of cation & $t_a =$ Transport Number of anion ### Solution & Colligative Properties - **Osmotic Pressure :** 1. $\pi = \rho gh$ - Where, $\rho =$ density of soln., $h =$ equilibrium height. 2. **Vont - Hoff Formula** (For calculation of O.P.) - $\pi = C R T$ - $\pi = C R T$ (just like ideal gas equation) - $C =$ total conc. of all types of particles. - $C = C_1 + C_2 + C_3 + ..........$ - $\pi = \frac{(n_1 + n_2 + n_3 + .........)}{V} RT$ - Note : If V$_1$ ml of C$_1$ conc. + V$_2$ ml of C$_2$ Conc. are mixed. - $\pi = \left(\frac{C_1 V_1 + C_2 V_2}{V_1 + V_2}\right) RT$ ; $\pi = \frac{(\pi_1 V_1 + \pi_2 V_2)}{RT}$ - **Type of solutions :** - (a) **Isotonic solution** - Two solutions having same O.P. - $\pi_1 = \pi_2$ (at same temp.) - (b) **Hyper tonic**- If $\pi_1 > \pi_2$. $\rightarrow$ 1st solution is hypertonic w.r.t. 2nd solution. - (c) **Hypotonic**- If $\pi_1 1 \Rightarrow$ dissociation. - $i P_B^\circ$ ; A is more volatile than B - B.P. of A (X_A P_A^\circ + X_B P_B^\circ)$ - (ii) A $\rightleftharpoons$ B $\rightarrow$ A----B - B $\rightleftharpoons$ B $\rightarrow$ B----B - Weaker force of attraction - (iii) $\Delta H_{\text{mix}} = +ve$ energy absorbed - (iv) $\Delta V_{\text{mix}} = +ve$ ($1L + 1L > 2L$) - (v) $\Delta S_{\text{mix}} = +ve$ - (vi) $\Delta G_{\text{mix}} = -ve$ - eg. H$_2$O + C$_2$H$_5$OH - H$_2$O + CH$_3$OH - H$_2$O + C$_2$H$_5$OH - C$_6$H$_6$ + C$_2$H$_5$OH - CCl$_4$ + C$_6$H$_6$ - CCl$_4$ + CH$_3$OH $\rightarrow$ dipole dipole interaction becomes weak. - (Diagram showing azeotropic mixture for positive deviation) - (b) **Negative deviation** - (i) $P_T^{\text{exp}} ### Solid State - **Classification of Crystal into Seven System** - | Crystal System | Unit Cell Dimensions and Angles | Bravais Lattices | Example | |---|---|---|---| | Cubic | $a=b=c; \alpha=\beta=\gamma=90^\circ$ | SC, BCC, FCC | NaCl | | Orthorhombic | $a \ne b \ne c; \alpha=\beta=\gamma=90^\circ$ | SC, BCC, end centred & FCC | S$_R$ | | Tetragonal | $a=b \ne c; \alpha=\beta=\gamma=90^\circ$ | SC, BCC | Sn, ZnO$_2$ | | Monoclinic | $a \ne b \ne c; \alpha=\gamma=90^\circ \ne \beta$ | SC, end centred | S$_W$ | | Rhombohedral | $a=b=c; \alpha=\beta=\gamma \ne 90^\circ$ | SC | Quartz | | Triclinic | $a \ne b \ne c; \alpha \ne \beta \ne \gamma \ne 90^\circ$ | SC | H$_3$BO$_3$ | | Hexagonal | $a=b \ne c; \alpha=\beta=90^\circ; \gamma=120^\circ$ | SC | Graphite | - **Analysis of Cubical System** - | Property | SC | BCC | FCC | |---|---|---|---| | (i) Atomic radius ($r$) | $a/2$ | $\sqrt{3}a/4$ | $a/(2\sqrt{2})$ or $a/2.28$ | | (ii) No. of atoms per unit cell ($Z$) | 1 | 2 | 4 | | (iii) C.No. | 6 | 8 | 12 | | (iv) Packing efficiency | 52% | 68% | 74% | | (v) No. voids | | | | | (a) Octahedral ($Z$) | -- | -- | 4 | | (b) Tetrahedral ($2Z$) | -- | -- | 8 | - **Neighbourhood of a particle :** - **Simple Cubic (SC) Structure :** - | Type of neighbour | Distance | no.of neighbours | |---|---|---| | nearest | $a$ | 6 (shared by 4 cubes) | | (next)$^1$ | $a/\sqrt{2}$ | 12 (shared by 2 cubes) | | (next)$^2$ | $a/\sqrt{3}$ | 8 (unshared) | - (ii) **Body Centered Cubic (BCC) Structure :** - | Type of neighbour | Distance | no.of neighbours | |---|---|---| | nearest | $2r = a \frac{\sqrt{3}}{2}$ | 8 | | (next)$^1$ | $a$ | 6 | | (next)$^2$ | $a\sqrt{2}$ | 12 | | (next)$^3$ | $a\sqrt{11}/2$ | 24 | | (next)$^4$ | $a\sqrt{3}$ | 8 | - (iii) **Face Centered Cubic (FCC) Structure :** - | Type of neighbour | Distance | no.of neighbours | |---|---|---| | nearest | $a / \sqrt{2}$ | 12 (= $3 \times 8 / 2$) | | (next)$^1$ | $a$ | 6 (= $3 \times 8 / 4$) | | (next)$^2$ | $a \sqrt{3/2}$ | 24 | | (next)$^3$ | $a \sqrt{2}$ | 12 | | (next)$^4$ | $a \sqrt{5/2}$ | 24 | - **Density of Lattice Matter (d) =** $\frac{Z M}{N_A a^3}$ where $N_A =$ Avogadro's No. M = atomic mass or molecular mass. - **Ionic Crystals** - | C.No. | Limiting radius ratio ($\frac{r_+}{r_-}$) | |---|---| | 3 | 0.155 - 0.225 (Triangular) | | 4 | 0.225 - 0.414 (Tetrahedral) | | 6 | 0.414 - 0.732 (Octahedral) | | 8 | 0.732 - 0.999 (Cubic). | - **Examples of Ionic Crystal** - (a) Rock Salt (NaCl) Coordination number (6 : 6) - (b) CsCl C.No. (8 : 8) - Edge length of unit cell - $a_{\text{w}} = \frac{2}{\sqrt{3}} (r_c + r_a)$ - (c) Zinc Blende (ZnS) C.No. (4 : 4) - $a_{\text{cs}} = \frac{4}{\sqrt{3}} (r_{\text{Zn}^{2+}} + r_{S^{2-}})$ - (d) Fluorite structure (CaF$_2$) C.No. (8 : 4) - $a_{\text{frc}} = \frac{4}{\sqrt{3}} (r_{\text{Ca}^{2+}} + r_{F^-})$ - **Crystal Defects (Imperfections)** - (Diagram classifying crystal defects into Stoichiometric and Non-Stoichiometric, and further into Schottky, Frenkel, Metal excess, Non-Metal excess, Electron in place of anion, extra cation in the interstitial, Vacant site in place of cation, extra anion in interstitial (not found)) ### Chemical Kinetics & Radioactivity - **Rate/Velocity of Chemical Reaction:** - Rate = $\frac{\Delta c}{\Delta t}$ mol/lit sec$^{-1}$ = mol lit$^{-1}$ time$^{-1}$ = mol dm$^{-3}$ time$^{-1}$ - **Types of Rates of chemical reaction :** - For a reaction R $\rightarrow$ P - Average rate = $\frac{\text{Total change in concentration}}{\text{Total time taken}}$ - $R_{\text{instantaneous}} = \lim_{\Delta t \to 0} \frac{\Delta c}{\Delta t} = \frac{dc}{dt} = \frac{d[R]}{dt} = \frac{d[P]}{dt}$ - (Graph of Concentration vs Time) - **Rate Law (Dependence of rate on concentration of reactants):** - **Rate = $K$ (conc.)$^{\text{order}}$** - differential rate equation or rate expression - Where K = Rate constant or specific reaction rate = rate of reaction when concentration is unity - unit of K = (conc)$^{1-\text{order}}$ time$^{-1}$ - **Order of reaction :** - Order of reaction = $a+m+b$ $\rightarrow$ products. - $R = [A]^a [B]^b$ Where $p$ may or may not be equal to $m$, & similarly $q$ may or may not be equal to $n$. - $p$ is order of reaction with respect to reactant A and $q$ is order of reaction with respect to reactant B and $(p+q)$ is overall order of the reaction. - **Integrated Rate Laws:** - $C_0$ or 'a' is initial concentration and $C_t$ or $a-x$ is concentration at time 't' - (a) **Zero order reactions :** - Rate = $k$ [conc.]$^0$ = constant - Rate = $k = \frac{C_0 - C_t}{t}$ or $C_t = C_0 - kt$ - Unit of K = mol lit$^{-1}$ sec$^{-1}$. Time for completion = $\frac{C_0}{k}$ - at $t_{1/2}$, $C_t = \frac{C_0}{2}$, so $kt_{1/2} = \frac{C_0}{2}$ $\Rightarrow$ $t_{1/2} = \frac{C_0}{2k}$ - $t_{1/2} \propto C_0$. - (b) **First Order Reactions :** - (i) Let a $1^{st}$ order reaction is, A $\rightarrow$ Products - $t = \frac{2.303}{k} \log \frac{a}{a-x}$ or $t = \frac{2.303}{k} \log \frac{C_0}{C_t}$ - $t_{1/2} = \frac{\ln 2}{k} = \frac{0.693}{k}$ = Independent of initial concentration. - $t_{\text{avg}} = \frac{1}{k} = 1.44 t_{1/2}$. - **Graphical Representation :** - $t = \frac{2.303}{k} \log C_0 - \frac{2.303}{k} \log C_t$ - (Graph of $\log C_t$ vs t for first order reaction) - (Graph of $\log C_t / C_0$ vs t for first order reaction) - (c) **Second order reaction :** - $2^{\text{nd}}$ order Reactions - Two types - A + A $\rightarrow$ products - $a$ $a$ $0$ - $(a-x)$ $(a-x)$ - $\frac{dx}{dt} = k (a-x)^2$ - $\Rightarrow \frac{1}{a-x} = kt + \frac{1}{a}$ - A + B $\rightarrow$ products - $a$ $b$ $0$ - $(a-x)$ $(b-x)$ - $\frac{dx}{dt} = k (a-x) (b-x)$ - $k = \frac{1}{t(a-b)} \log \frac{b(a-x)}{a(b-x)}$ - (d) **Psuedo first order reaction :** - For A + B $\rightarrow$ Products - [Rate = K $[A][B]$] - Now if 'B' is taken in large excess $b >> a$. - $\Rightarrow k = \frac{2.303}{t} \log \frac{a}{a-x}$ - 'b' is very large can be taken as constant - $\Rightarrow k'b = \frac{2.303}{t} \log \frac{a}{a-x}$ $\Rightarrow k' = \frac{2.303}{t} \log \frac{a}{a-x}$ , $K'$ is psuedo first order rate constant - **Methods to determine order of a reaction** - (a) **Initial rate method :** - If $R = k [A]^p [B]^q [C]^r$ - $[A] = \text{constant}$ - $[B] = \text{constant}$ - $[C] = \text{constant}$ - then for two different initial concentrations of A we have - $R_{0_1} = k [A_1]^p$ , $R_{0_2} = k [A_2]^p$ $\Rightarrow \frac{R_{0_1}}{R_{0_2}} = \left(\frac{[A_1]}{[A_2]}\right)^p$ - (b) **Using integrated rate law :** It is method of trial and error. - (c) **Method of half lives :** - $t_{1/2} = \frac{1}{k R_0^{n-1}}$ for $n^{th}$ order reaction - (d) **Ostwald Isolation Method :** - Methods to monitor the progress of the reaction : - (a) **Progress of gaseous reaction can be monitored by measuring total pressure at a fixed volume & temperature** - $P_0 (\Delta n - 1)$ - $n P_0 - P_{\text{total}}$ (Formula is not applicable when $n=1$, the value of $n$ can be fractional also.) - $P_{\text{total}} = P_0 + (n-1)x$ - $x = (P_{\text{total}} - P_0) / (n-1)$ - $P_A = P_0 - x = P_0 - (P_{\text{total}} - P_0) / (n-1)$ - $k = \frac{2.303}{t} \log \frac{P_0}{P_0 - x}$ - (b) **By titration method :** - 1. $a = V_t$ - $a-x = V_0$, $\Rightarrow k = \frac{2.303}{t} \log \frac{V_t}{V_0}$ - 2. Study of acid hydrolysis of an easter. - $k = \frac{2.303}{t} \log \frac{V_\infty - V_t}{V_\infty - V_0}$ - (c) **By measuring optical rotation produced by the reaction mixture :** - $k = \frac{2.303}{t} \log \frac{\theta_0 - \theta_\infty}{\theta_t - \theta_\infty}$ - **Effect of Temperature on Rate of Reaction.** - T.C. = $\frac{K_{t+10}}{K_t}$ = 2 to 3 (for most of the reactions) - **Arrhenius theory of reaction rate.** - (Diagram showing energy profile for exothermic and endothermic reactions with activation energy) - $E_{r_1} > E_{r_2}$ $\rightarrow$ endothermic - $E_{r_1} 0$ - $\ln K = \ln A - \frac{E_a}{RT}$ - $T \rightarrow \infty$, $K \rightarrow A$. - **Reversible Reactions** - $K_f = A_f e^{-E_f/RT}$ - $K_b = A_b e^{-E_b/RT}$ - $K_{\text{eq}} = \frac{K_f}{K_b} = \frac{A_f}{A_b} e^{-(E_f - E_b)/RT} = \frac{A_f}{A_b} e^{-\Delta H/RT}$ - (ii) **Reversible $1^{st}$ order Reaqation ( both forward and backward)** - $x = \frac{K_1 a}{K_1 + K_2} (1 - e^{-(K_1 + K_2)t})$ - $K_1 + K_2 = \frac{1}{t} \ln \frac{x_{\text{eq}}}{x_{\text{eq}} - x}$ - (iii) **Sequential $1^{st}$ order reaction** - $[A] = [A]_0 e^{-k_1 t}$ - $x = a (1 - e^{-k_1 t})$ - $y = \frac{K_1 a}{K_2 - K_1} (e^{-k_1 t} - e^{-k_2 t})$ - $t_{\text{max}} = \frac{1}{K_2 - K_1} \ln \frac{K_2}{K_1}$ - (Graph of Conc vs t for sequential first order reactions for Case I: $K_1 >> K_2$ and Case II: $K_2 >> K_1$) ### Inorganic Chemistry #### Periodic Table & Periodicity - **Development of Modern Periodic Table:** - (a) **Dobereiner's Triads:** He arranged similar elements in the groups of three elements called as triads - (b) **Newland's Law of Octave:** He was the first to correlate the chemical properties of the elements with their atomic masses. - (c) **Lother Meyer's Classification :** He plotted a graph between atomic masses against their respective atomic volumes for a number of elements. He found the observations : (i) elements with similar properties occupied similar positions on the curve. (ii) alkali metals having larger atomic volumes occupied the crests. (iii) transition elements occupied the troughs. (iv) the halogens occupied the ascending portions of the curve before the inert gases and - (v) alkaline earth metals occupied the positions at about the mid points of the ascending portions of the curve. On the basis of these observations he concluded that the atomic volumes (a physical property) of the elements are the periodic function of their atomic masses. - (d) **Mendeleev's Periodic Table:** - (e) **Mendeleev's Periodic's Law** - the physical and chemical properties of the elements are the periodic functions of their atomic masses. - | Periods | Number of Elements | Called as | |---|---|---| | (1) $n=1$ | 2 | Very Short period | | (2) $n=2$ | 8 | Short period | | (3) $n=3$ | 8 | Short period | | (4) $n=4$ | 18 | Long period | | (5) $n=5$ | 18 | Long period | | (6) $n=6$ | 32 | Very Long period | | (7) $n=7$ | 19 | Incomplete period | - **Merits of Mendeleev's Periodic table:** - It has simplified and systematised the study of elements and their compounds. - It has helped in predicting the discovery of new elements on the basis of the blank spaces given in its periodic table. - **Demerits in Mendeleev's Periodic table:** - Position of hydrogen is uncertain. It has been placed in IA and VIIA groups - No separate positions were given to isotopes. - Anomalous positions of lanthanides and actinides in periodic table. - Order of